The non-tempered theta 10 Arthur parameter and Gross-Prasad Conjectures

TL;DR

This paper constructs non-tempered Arthur packets for SO(3,2) and SO(4,1), investigates their restriction properties, and reveals limitations of the Gross-Prasad conjecture in non-generic cases, using theta correspondence techniques.

Contribution

It introduces explicit constructions of non-tempered Arthur packets for specific groups and analyzes their restriction behavior, highlighting new phenomena beyond generic cases.

Findings

Non-generic A-packets do not satisfy naive Gross-Prasad conjecture.

Established multiplicity formulas for constructed packets.

Demonstrated non-vanishing of automorphic periods in certain cases.

Abstract

We provide a construction of local and automorphic non-tempered Arthur packets of the group SO(3,2) and its inner form SO(4,1) associated with a certain Arthur's parameter and prove a multiplicity formula. We further study the restriction of the representations in these packets to the subgroup SO(3,1). In particular, we discover that the local Gross-Prasad conjecture, formulated for generic L-packets, does not generalize naively to a non-generic A-packet. We also study the non-vanishing of the automorphic SO(3,1)-period on the group SO(4,1) x SO(3,1) and SO(3,2) x SO(3,1) for the representations above. The main tool is the local and global theta correspondence for unitary quaternionic similitude dual pairs.